Earth-related coordinates of spatial data can be either

• projected: measured on a two-dimensional surface, which is associated with a projection from the Earth’s surface
• geocentric: x-, y-, and z-coordinates associated with the three principal directions, measured from Earth’s center
• geographic: degrees latitude and longitude, associated with a datum (ellipsoidal or even spherical model of the Earth)

Geometrical computations on projected or geocentric coordinates can be done using Euclidean geometry, where all lines are staight.

Spherical (or ellipsoidal) geometry computes distances, direction, areas etc. over the surface of the sphere (ellipsoid).

• I believe that applying Eucledian geometry to geographical coordinates is the number one most common error made in spatial data science (closely followed by ignoring the support of data).

• technological advances have made spherical geometry a good option, but inertia in legacy GIS and data science languages have slowed down its adoption / uptake

• We have gotten used to Plate Carree, and are OK with it.

• Even GeoJSON has written down that the world is 2D.

https://xkcd.com/977/

https://en.wikipedia.org/wiki/Equirectangular_projection

Compared to other projections:

• every global projection is miserable in its own way
• it has true scale (1 unit easting equals 1 unit Northing) only at the equator
• it is hopeless at the poles and at the antimeridian

Compared to an ellipsoid, or sphere:

• It is 2D. The world is round, bro.

Consider, for a moment, the difference between:

• a flat projection of the sphere, and a sphere (1, 2)
• a sphere and an ellipsoid with 1/300 flattening (2, 3)

Equirectangular projection: where meridians and parallels form equal rectangles (left; true scale at centre), not squares (right: nowhere true)

• GeographicLib (C.F.F. Karney), “[…] a small set of C++ classes for […] solving geodesic problems.”
  • https://geographiclib.sourceforge.io/
  • exact up to 8-byte doubles
  • part of Proj.4, now PROJ
• PostGIS (liblwgeom)’s “geograpy” type
  • funded by palantir
  • feature list,
  • but: “The buffer and intersection functions are actually wrappers on top of a cast to geometry, and are not carried out natively in spherical coordinates. As a result, they may fail to return correct results for objects with very large extents that cannot be cleanly converted to a planar representation.”
• R package geosphere providing distance, bearing, etc, and several alternative measures
• R packages sp, gstat, spdep using geodetic distances in case of geographic coordinates
• R package lwgeom interfacing liblwgeom

• in contrast to the plane \(R^2\), the sphere (\(S^2\)) is bounded
• where in \(R^2\) polygons have an unambiguous inside, on \(S^2\) polygons divide the sphere’s surface in two parts.
• the inside can then e.g. be determined by ring direction (left of the boundary, when traversing the ring), and flips if the ring directions is reversed
• the empty polygon has a complement: the full polygon.
• two new bounding structures:
  • bounding cap: smalles circumfering circle (center point + radius)
  • bounding rectangle: lat/lng range that might cross the antimeridian

• Google Earth
• Google Maps: now has “enable/disable globe view” switches between azimuthal perspective and Web Mercator.
• Cesium (can switch between WM, perspective and orthographic)

Measures

distance, area, direction, …, bounding box, bounding cap

Predicates

intersects, overlaps, covers, touches, is_within_distance, …

Transformations

intersection, union, difference, sym_difference, buffer

• an open source library written and supported by Google, powering
  • Google maps, earth, earth engine
  • Google’s serverless SQL engine bigquery GIS
• provides all the measures, predicates, transformations
• allows several snapping to grid options, preserving validity
• provides, and uses, a spatial index by using a space-filling hilbert curve, after projecting a sphere onto six cube-faces of an earth-cube

https://s2geometry.io/devguide/s2cell_hierarchy.html

References

• Karney, Charles F.F. 2013. “Algorithms for Geodesics.” Journal of Geodesy 87 (1). Springer: 43–55. https://link.springer.com/content/pdf/10.1007/s00190-012-0578-z.pdf.

• R. Hijmans, 2020, Introduction to the geosphere package; CRAN package vignette, https://cran.r-project.org/web/packages/geosphere/vignettes/geosphere.pdf

• E Pebesma, D Dunnington, Spherical geometry in sf using s2geometry, https://r-spatial.github.io/sf/articles/sf7.html